5. Viscosity Solutions and Large-Time Behavior for Non-Resonant Balance Laws

Our aim is now to summarize several rigorous results concerning BV solutions of general hyperbolic systems of balance laws with source terms about which no dissipation hypothesis is made. Results of this kind owe to the pioneering 1979 paper by Tai-Ping Liu [38], already quoted in Chapter 4, inside which an astute extension of the Glimm scheme [21] is studied in great detail. His existence theorems can be somewhat simplified when source terms are concentrated inside the whole Riemann fan instead of being localized in the middle of computational cells: hence the presentation adopted in [1] will be preferred. Uniqueness results were also proved in that more recent paper (a slightly less complete result was obtained in [24]). A second part in the chapter focuses onto decay results: first, positive (rarefaction) waves are handled following the paper [23]. Second, weak and strong decay results dealing with time-asymptotic behavior are recalled from [38]: as a consequence of the interaction potential’s decay, the BV solution endowed with constant states at infinity evolves toward a non-interacting wave pattern, which resembles more and more to the fan associated to the Riemann problem resolving that discontinuity. Inside the area |x| ≤ M where the source term is powerful, a steady-state profile, smooth solution of ∂ _x f (u) = g(x, u) emerges as time passes whereas in the complementary domain, only homogeneous hyperbolic waves survive. In other words, this is exactly the setup for scattering theory, namely a wave phenomenon studied in a domain which is large compared to the characteristic scale of the interaction. The non-resonance assumption |∇f (u)| ≠ 0 is of paramount importance here because it ensures that convective waves never stagnate and exit the interval |x| ≤ M in finite time thus the solution cannot be amplified in an unbounded manner by the source term.